ORIGINAL_ARTICLE
Directed prime graph of non-commutative ring
Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$. Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.
https://as.yazd.ac.ir/article_1963_e3b1632e24eaf910a4e5c59d08e3c7ce.pdf
2021-02-01
1
12
10.22034/as.2021.1963
Directed Graph
Non-commutative Ring
Prime graph
Sanjoy
Kalita
sanjoykalita1@gmail.com
1
Department of Mathematics, Gauhati University, Guwahati- 781014, Assam, India
LEAD_AUTHOR
Kuntala
Patra
kuntalapatra@gmail.com
2
Department of Mathematics, Gauhati University, Guwahati- 781014, Assam, India
AUTHOR
[1] S. Akbari, A. Mohammadian , On The Zero-Divisor Graph of A Commutative Ring, J. Algebra , Vol 274 (2004), 847-855.
1
[2] S. Akbari, A. Mohammadian, Zero-Divisor Graphs of Non-Commutative Rings, J. Algebra, Vol 296 (2006), 462-479.
2
[3] D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J.Algebra 217 (1999), No. 2, 434 - 447.
3
[4] D. F. Anderson, S. B. Mulay, The Diameter and Girth of A Zero-Divisor Graph, J. Pure App. Algebra, Vol 210(2007), No 2, 543550 .
4
[5] S. E. Atani, A. Y. Darani , Zero-Divisor Graphs with respect to Ideals in Non-commutative Rings, ISRN Discrete Math., Vol (2011) (Article ID 459547, 7 pages, doi:10.5402/2011/459547)
5
[6] I. Beck , Coloring of commutative rings, J. Algebra 116 (1988) No. 1, 208 - 226.
6
[7] I. Bozic, Z. Petrovic, Zero-Divisor Graphs Of Matrices Over Commutative Rings, Comm. Alg. Vol 37 (2009), 1186 -1192.
7
[8] F. Harary, Graph Theory, Eddison Wesley Publishing Company inc. 1969.
8
[9] J. Lambek, Lectures on Rings and Modules , Blaisdel Publ. Co., 1966.
9
[10] T. G. Lucas , The Diameter Of A Zero Divisor Graph, J. Algebra, Vol 301(2006), 174193.
10
[11] H. R. Maimani, M. R. Pournaki, S. Yessemi, Zero Divisor Graph with respect to An Ideal, Comm. Alg., Vol. 34(2006), 923929.
11
[12] K. K. Rajkhowa, H. K. Saikia , Zero-Divisor Graphs Of Non-Commutative Rings, Adv. Appl. Discrete Math., Vol. 10 (2012), No. 1, 49- 64.
12
[13] S. P. Redmond, The Zero-Divisor Graph Of A Non-Commutative Ring, International J. Commutative Rings, Vol 1(4) (2002), 203 211.
13
[14] Bh. Satyanarayana, K. Shyam Prasad, D. Nagaraju, Prime Graph of a Ring, J. of Combinatorics, Information and System Sources, Vol 35 (2010) No 1-2, 27-42.
14
ORIGINAL_ARTICLE
Limits and colimits in the category of pre-directed complete pre-ordered sets
In this paper, some categorical properties of the category { Pre-Dcpo} of all pre-dcpos; pre-ordered sets which are also pre-directed complete, with pre-continuous maps between them is considered. In particular, we characterize products and coproducts in this category. Furthermore, we show that this category is neither complete nor cocomplete. Also, epimorphisms and monomorphisms in {Pre-Dcpo} are described.Finally, some adjoint relations between the category {Pre-Dcpo} and others are considered.More precisely, we consider the forgetful functors between this category and some well-known categories, and study the existence of their left and right adjoints.
https://as.yazd.ac.ir/article_1833_1cd9994a1851d286fea6142efe53f185.pdf
2021-02-01
13
23
10.22034/as.2020.1833
Cofree
Coproduct
Free
Pre-dcpo
Product
Halimeh
Moghbeli
h_moghbeli@sbu.ac.ir
1
Depatment of Mathematcs, Faculty of science, University of Jiroft, Jiroft, Iran
LEAD_AUTHOR
[1] S. Abramsky and A. Jung, Domain theory. In: Handbook of Computer Science and logic, vol. 3, Clarendon Press, Oxford, (1995).
1
[2] J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Categories. The Joy of Cats, http://katmat.math.uni-bremen.de/acc/acc.pdf
2
[3] A. Fiech, Colimits in the category DCPO, Math. Structures Comput. Sci. Vol. 6 (1996), pp. 455-468.
3
[4] A. Jung, Cartesian closed categories of algebraic cpos, Theoret. Comp. Sci. Vol. 70 (1990), pp. 233-250.
4
[5] M. Mahmoudi, H. Moghbeli and K. Pi oro, Natural congruences and isomorphism theorem for directed complete posets, Algebra Universalis. Vol. 77 No. 1 (2017), pp. 79-99, DOI: 10.1007/s00012-017-0424-5.
5
[6] M. Mahmoudi, H. Moghbeli and K. Pi oro, Directed complete poset congruences, Journal of Pure and Applied Algebra. Vol. 223 No. 10 (2019), pp. 4161-4170.
6
[7] S. Bulman-Fleming and M. Mahmoudi, The category of S-posets, Semigroup Forum. Vol. 71 No. 3 (2005), pp. 443-461.
7
[8] R. L. Crole, Categories for types, Cambridge University Press, Cambridge, (1994).
8
[9] S. Vickers and C. Townsend, A universal characterization of the double powerlocale, Theoretical Computer Science. Vol. 316 (2004), pp. 297-321.
9
[10] D. S. Scott, Domains for denotational semantics, In Nielsen, M., Schmidt, E.M., eds.: Automata, Languages and Programming, Proceedings of the 9th Colloquium, July 1982, Aarhus, Denmark, Lect. Notes Comp. Sci. Vol. 140 (1982), pp. 577-613.
10
ORIGINAL_ARTICLE
On $\mathbb{Z}G$-clean rings
Let $R$ be an associative ring with unity. An element $x \in R$ is called $\mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $\mathbb{Z}G$-regular element in $R$. A ring $R$ is called $\mathbb{Z}G$-clean if every element of $R$ is $\mathbb{Z}G$-clean. In this paper, we show that in an abelian $\mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $\frac{R}{Nil(R)}$ is $G$-regular. Furthermore, we characterize $\mathbb{Z}G$-clean rings. Also, this paper is involved with investigating $\mathbb{F}_{2}C_{2}$ as a social group and measuring influence a member of it’s rather than others.
https://as.yazd.ac.ir/article_1834_67678c88da5df6a3fde922d7c1091103.pdf
2021-02-01
25
40
10.22034/as.2020.1834
Social group
Strongly ZG-regular
Von Neumann regular
ZG-clean
ZG-regular
Marzieh
Farmani
mino.farmani@riau.ac.ir
1
Islamic Azad university, Roudehen branch, Roudehen, Iran.
LEAD_AUTHOR
[1] D. D. Anderson, P. V. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra, 30 (2002), 3327-3336.
1
[2] N. Ashra, E. Nasibi, On r-clean rings, Mathematical Reports 2013; 15 (65): No. 2.
2
[3] G. Azumaya, Strongly -regular rings, J. Fac. Sci. Hokkaido Univ. 13 (1954), 34-39.
3
[4] P. Ara, -Regular rings have stable range one, Proc. Amer. Math. Soc., 124 (11) (1996), 3293-3298.
4
[5] A. Badawi, On semicommutative -regular rings, Commun. Algebra, 22 (1) (1993), 151-157.
5
[6] A. Badawi, Abelian -regular rings, Commun. Algebra, 25 (4) (1997), 1009-1021.
6
[7] P. V. Camillo, H. P. Hu, Exchange rings, Unit and idempotents, Comm. Algebra. 22 (1994), 4737-4749.
7
[8] R. Yue Chi Ming, On Von Neumann regular rings , III, Monatshefte fur Mathematik, 86 (1978), 251-257.
8
[9] P. Danchev, J. Ster,Generalizing -regular ring, Taiwanese J. Math, 19 (6) (2014), 1577-1592.
9
[10] M. F. Dischinger, Sur les anneaux fortment -reguliers, C. R. Acad. Sci.Paris, Ser. A 283 (1976), 571-573.
10
[11] K. R. Goodearl, Von Neumann regular rings, Monographs and studied in Math. 4, Pitman, London (1979).
11
[12] D. Handelman D, Perspectivity and cancellation in regular rings, J. Algebra, 48 (1977), 1-16.
12
[13] Ch. Huh, N. K. Kim, Y. Lee, Example of strongly -regular rings, Pure Appl. Algera, 189 (2004), 195-210.
13
[14] Nicholas A. Immormino, Clean rings clean group rings, A Ph.D Dissertation, the Graduate College of Bowling Green, (2013).
14
[15] T.Y. Lam, A rst course in noncommutative rings, Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, (2001).
15
[16] W. K. Nicholson, Clean rings: A Survey, Advances in Ring Theory. Hackensack, NJ: World Sci. Publ., (2005), 181-198.
16
[17] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269-278.
17
[18] Sh. Safari Sabet, M. Farmani, Extensions of Regular Rings, International Journal of Industrial Mathematics, 8 (4) (2016), 331-337.
18
[19] Sh. A. Safari Sabet, M. Karami, G-regular and strongly G-regular Rings, IJONS. 26 (5), (2014) 1953-1958.
19
ORIGINAL_ARTICLE
Boolean expression based on hypergraphs with algorithm
This paper introduces a novel concept of Boolean function--based hypergraph with respect to any given T.B.T(total binary truth table). This study defines a notation of kernel set on switching functions and proves that every T.B.T corresponds to a Minimum Boolean expression via kernel set and presents some conditions on T.B.T to obtain a Minimum irreducible Boolean expression from switching functions. Finally, we present an algorithm and so Python programming(with complete and original codes) such that for any given T.B.T, introduces a Minimum irreducible switching expression.
https://as.yazd.ac.ir/article_1835_d021d23b138f47c633f5d0f40c1ae173.pdf
2021-02-01
41
60
10.22034/as.2020.1835
Boolean function--based hypergraph
Switching function
switching kernel
T.B.T
Mohammad
Hamidi
m.hamidi@pnu.ac.ir
1
Department of mathematics, Payame Noor university, Tehran, Iran.
LEAD_AUTHOR
Marzieh
Rahmati
m.rahmati@pnu.ac.ir
2
Department of mathematics, Payame Noor university, Tehran, Iran.
AUTHOR
Akbar
Rezaei
rezaei@pnu.ac.ir
3
Department of Mathemtics, Payame Noor University, Tehran, Iran
AUTHOR
[1] C. Berge, Graphs and Hypergraphs, North Holland, 1979.
1
[2] P. Chandra R. K. Singh, Y., Generation of mutants for boolean expression, J. Discrete Math. Sci. Cryptogr., (2014), pp. 589-607.
2
[3] J. Eldon Whitesitt, Boolean Algebra and Its Applications, New York Dover Publications, Inc., 1995.
3
[4] M. Hamidi, A. Borumand saied, Accessible single-valued neutrosophic graphs, J. Appl. Math. Comput., Vol. 57 (2018), pp. 121-146.
4
[5] M. Hamidi, A. Borumand saied, Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks, New Math. Nat. Comput., Vol. 14 No. 2 (2018), pp. 157-185.
5
[6] M. Hamidi and A. Borumand Saeid, On Derivable Trees, Trans. Combin., Vol. 8 N. 2 (2019), pp. 21-43.
6
[7] M. Hamidi and F. Smarandache, Single-Valued Neutrosophic Directed (Hyper)Graphs And Applications in Networks, J. Intell. Fuzzy. Syst., Vol. 37 N. 2 (2019), pp. 2869-2885.
7
[8] M. Hamidi and R. Ameri, -Derivable Digraphs and its Application in Wireless Sensor Networking, Discrete. Math. Algorithms. Appl., (2020).
8
[9] B. Molnar, Applications of Hypergraphs in Informatics a Survey and Opportunities for Research, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 42 (2014), pp. 261-282.
9
[10] F. Smarandache, Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra, Neutrosophic Sets
10
and Systems, Vol. 33 (2020), pp. 290-296.
11
[11] F. Smarandache, n-SuperHyperGraph and Plithogenic n-SuperHyperGraph, in Nidus Idearum, Vol. 7, second
12
edition, Pons asbl, Bruxelles, pp. 107-113, 2019; http://fs.unm.edu/NidusIdearum7-ed2.pdf.
13
ORIGINAL_ARTICLE
$r$-Submodules and $uz$-modules
In this article we study and investigate the behavior of $r$-submodules (a proper submodule $N$ of an $R$-module $M$ in which $am\in N$ with ${\rm Ann}_M(a)=(0)$ implies that $m\in N$ for each $a\in R$ and $m\in M$). We show that every simple submodule, direct summand, divisible submodule, torsion submodule and the socle of a module is an $r$-submodule and if $R$ is a domain, then the singular submodule is an $r$-submodule. We also introduce the concepts of $uz$-module (i.e., an $R$-module $M$ such that either ${\rm Ann}_M(a)\not=(0)$ or $aM=M$, for every $a\in R$) and strongly $uz$-module (i.e., an $R$-module $M$ such that $aM\subseteq a^2M$, for every $a\in R$) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly $uz$-module and every Artinian $R$-module is a $uz$-module. It is observed that if $M$ is a faithful cyclic $R$-module, then $M$ is a $uz$-module if and only if every its cyclic submodule is an $r$-submodule. In addition, in this case, $R$ is a domain if and only if the only $r$-submodule of $M$ is zero submodule. Finally, we prove that $R$ is a $uz$-ring if and only if every faithful cyclic $R$-module is a $uz$-module.
https://as.yazd.ac.ir/article_1858_1e18b72399241ed905159156681c9c6e.pdf
2021-02-01
61
73
10.22034/as.2020.1858
$r$-ideal
$r$-submodule
strongly $uz$-module
$uz$-module
Rostam
Mohamadian
mohamadian_r@scu.ac.ir
1
Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
LEAD_AUTHOR
[1] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, Berlin/Heidelberg, New York, 1992.
1
[2] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesely, Reading Mass,1969.
2
[3] R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker Inc, New York, 1972.
3
[4] R. Gilmer, W. Heinzer, On Jonsson modules over a commutative ring, Acta Sci. Math. 46 (1983), 3-15.
4
[5] J. A. Huckabo, Commutative Ring with Zero Divisors, Marcel Dekker Inc, 1988.
5
[6] T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, Berlin/Heidelberg,New York, 1989.
6
[7] I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
7
[8] O. A. S. Karamzadeh, Module whose countably generated submodules are epimorphic images, Colloq. Math. 46 (1982), 143-146.
8
[9] S. Koc, U. Tekir, r-Submodules and special r-Submodules, Turk. J. Math. 42 (2018), 1863-1876.
9
[10] T. Y. Lam, Lecture on Modules and Rings, Springer, 1999.
10
[11] J. Lambek, Lecture on Rings and Modules, Waltham-Toronto-London: Blaisdell, 1966.
11
[12] R. Mohamadian, r-ideals in commutative rings, Turk. J. Math. 39 (2015), 733-749.
12
[13] R. Y. Sharp, Steps in Commutative Algebra, Cambridge university press, 1990.
13
ORIGINAL_ARTICLE
Generalized stone residuated lattices
This paper introduces and investigates the notion of a generalized Stone residuated lattice. It is observed that a residuated lattice is generalized Stone if and only if it is quasicomplemented and normal. Also, it is proved that a finite residuated lattice is generalized Stone if and only if it is normal. A characterization for generalized Stone residuated lattices is given by means of the new notion of $\alpha$-filters. Finally, it is shown that each non-unit element of a directly indecomposable generalized Stone residuated lattice is a dense element.
https://as.yazd.ac.ir/article_1885_cdc1ba4f87c31ac8b6bac5bd6c4407cb.pdf
2021-02-01
75
87
10.22034/as.2020.1885
generalized Stone residuated lattice
normal residuated lattice
quasicomplemented residuated lattice
residuated lattice
Saeed
Rasouli
srasouli@pgu.ac.ir
1
Department of Mathematics, College of science, Persian Gulf University, Bushehr, 7516913817, Iran
LEAD_AUTHOR
[1] G. Birkho, Lattice theory, Vol. 25, American Mathematical Soc. (1940).
1
[2] C. C. Chang, Algebraic analysis of many valued logics, Transactions of the American Mathematical society Vol. 88 No. 2 (1958), pp. 467-490.
2
[3] R. Cignoli and A. T. Torrell, Boolean products of MV-algebras: hypernormal MV-algebras, Journal of mathematical analysis and applications Vol. 199 No. 3 (1996), pp. 637-653.
3
[4] L. C. Ciungu, Classes of residuated lattices, Annals of the University of Craiova Mathematics and Computer Science Series Vol. 33 (2006), pp. 189-207.
4
[5] W. H. Cornish, Normal lattices, Journal of the Australian Mathematical Society Vol. 14 No. 2 (1972), pp.200-215.
5
[6] W. H. Cornish, Annulets and -ideals in a distributive lattice, Journal of the Australian Mathematical Society Vol. 15 No. 1 (1973), pp. 70-77.
6
[7] A. Di Nola and S. Sessa, On MV-algebras of continuous functions, in Non-classical Logics and Their Applications to fuzzy Subsets, Springer, (1995), pp. 23-32.
7
[8] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated lattices: an algebraic glimpse at substructural logics, Vol. 151, Elsevier (2007).
8
[9] L. Gillman and M. Jerison, Rings of continuous functions, Courier Dover Publications (2017).
9
[10] G. Gratzer, Lattice theory: Foundation, Springer Science & Business Media (2011).
10
[11] G. Gratzer and E. T. Schmidt, On a problem of mh stone, Acta Mathematica Hungarica Vol. 8 No. 3-4 (1957), pp. 455-460.
11
[12] T. Katrinak, Remarks on stone lattices I,(russian) math, Fyz. Casopis Vol. 16 (1966), pp. 128.
12
[13] S. Rasouli, Generalized co-annihilators in residuated lattices, Annals of the University of Craiova Mathematics and Computer Science Series Vol. 45 No. 2 (2018a), pp. 190-207.
13
[14] S. Rasouli, Heyting, Boolean and Pseudo-MV lters in residuated lattices, J. Multiple Valued Log. Soft Comput. Vol 31 No 4 (2018b), pp. 287-322.
14
[15] S. Rasouli, The going-up and going-down theorems in residuated lattices, Soft Computing Vol. 23 No. 17 (2019), pp. 7621-7635.
15
[16] S. Rasouli, Quasicomplemented residuated lattices, Soft Computing Vol. 24 (2020), pp. 6591-6602.
16
[17] S. Rasouli and B. Davvaz, An investigation on boolean prime lters in BL-algebras, Soft Computing Vol. 19 No. 10 (2015), pp. 2743-2750.
17
[18] S. Rasouli and A. Dehghani, Topological residuated lattices, Soft Computing Vol. 24 No. 5 (2020), pp. 3179-3192.
18
[19] S. Rasouli and M. Kondo, n-normal residuated lattices, Soft Computing Vol. 24 No. 1 (2020), pp. 247-258.
19
[20] S. Rasouli and A. Radfar, PMTL lters, Rl lters and PBL lters in residuated lattices, Journal of Multiple Valued Logic and Soft Computing Vol. 29 No. 6 (2017), pp. 551-576.
20
[21] H. P. Sankappanavar and S. Burris, A course in universal algebra, Graduate Texts Math 78 (1981).
21
[22] T. Speed, Some remarks on a class of distributive lattices, Journal of the Australian Mathematical Society Vol. 9 No. 3-4 (1969a), pp. 289-296.
22
[23] T. Speed, Two congruences on distributive lattices, Bulletin de la Societe Royale des Sciences de Liege (1969b).
23
[24] J. Varlet, Contribution a l'etude des treillis pseudocomplementes et des treillis de Stone, Siege de la societe:Universite (1963).
24
[25] J. Varlet, A generalization of the notion of pseudo-complementedness, Bull. Soc. Roy. Sci. Liege Vol. 37 (1968), pp. 149-158.
25
ORIGINAL_ARTICLE
Modules whose nonzero finitely generated submodules are dense
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. First, we study multiplication $R$-modules $M$ where $R$ is a one dimensional Noetherian ring or $M$ is a finitely generated $R$-module. In fact, it is proved that if $M$ is a multiplication $R$-module over a one dimensional Noetherian ring $R$, then $M\cong I$ for some invertible ideal $I$ of $R$ or $M$ is cyclic. Also, a multiplication $R$-module $M$ is finitely generated if and only if $M$ contains a finitely generated submodule $N$ such that $Ann_R(N)= Ann_R(M)$. A submodule $N$ of $M$ is called dense in $M$, if $M=\sum_\varphi\varphi(N)$ where $\varphi$ runs over all the $R$-homomorphisms from $N$ into $M$ and $R$-module $M$ is called a weak $\pi$-module if every non-zero finitely generated submodule is dense in $M$. It is shown that a faithful multiplication module over an integral domain $R$ is a weak $\pi$-module if and only if it is a Prufer prime module.
https://as.yazd.ac.ir/article_1908_5372dd724763961313da714afdbf522a.pdf
2021-02-01
89
97
10.22034/as.2020.1908
Dense submodules
Multiplication modules
Prime modules
Weak $pi$-modules
Alireza
Hajikarimi
a.hajikarimi@mau.ac.ir
1
Department of Mathematics, Mobarakeh Branch, Islamic Azad University, Isfahan, Iran,
LEAD_AUTHOR
[1] M.M. Ali and D.J. Smith, Some remarks on multiplication and projective modules, Communications in Algebra, Vol. 32 No. 10 (2004), pp. 3897–3909.
1
[2] M. Alkan and Y. Tras, On invertible and dense submodules, Communications in Algebra, Vol. 32 No. 10 (2004), pp. 3911–3919.
2
[3] A. Azizi, Principal ideal multiplication modules, Algebra Colloquium, Vol. 15 No. 04 (2008), pp. 637–648.
3
[4] A. Barnard, Multiplication modules, Journal of Algebra, Vol. 71 (1981), pp. 174–178.
4
[5] M. Behboodi, On prime modules and dense submodules, Journal of commutative Algebra, Vol. 4 No. 4 (2012), pp. 479–488.
5
[6] Z.A. El-Bast and P.F. Smith, Multiplication modules, Communications in Algebra, Vol. 16 (1988), pp. 755–799.
6
[7] G.D. Findlay and J. Lambek, A generalized ring of quotients, I, II, Canadian Mathematical Bulletin, Vol. 1 (1958), pp. 77–85, 155–167.
7
[8] A. Hajikarimi and A.R. Naghipour, A note on monoform modules, Bulletin of the Korean Mathematical Society, Vol. 56 No. 2 (2019), pp. 505–514.
8
[9] A.G. Naoum and F.G. Al-Alwan, Dense submodules of multiplication modules, Communications in Algebra, Vol. 24 No. 2 (1996), pp. 413–424.
9
[10] A.G. Naoum and F.G. Al-Alwan, Dedekind modules, Communications in Algebra, Vol. 24 No. 2 (1996), pp. 397–412.
10
[11] W.K. Nicholson, J.K. Park and M.F. Yousif, Principally quasi-injective modules, Communications in Algebra, Vol. 27 No. 4 (1999), pp. 1683–1693.
11
[12] P.F. Smith, Some remarks on multiplication modules, Archiv der Mathematik, Vol. 50 (1988), pp. 223–235.
12
ORIGINAL_ARTICLE
Some classical theorems in state residuated lattices
This paper, by considering the notion of a state residuated lattice morphism in the class of state residuated lattices, investigates some classical theorems namely the going up and lying over theorems. Results show that each state residuated lattice morphism fulfills these theorems. Also, some properties about prime filters of residuated lattices are obtained which are given in the paper.
https://as.yazd.ac.ir/article_1910_495f3d096db2168cb40cf4441545ff60.pdf
2021-02-01
99
116
10.22034/as.2020.1910
going up theorem
lying over theorem
state residuated lattice
Mohammad
Taheri
taheri.mohamad96@yahoo.com
1
Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.
AUTHOR
Farhad
Khaksar Haghani
haghani1351@yahoo.com
2
Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.
LEAD_AUTHOR
Saeed
Rasouli
srasouli@pgu.ac.ir
3
Department of Mathematics, Persian Gulf University, Bushehr, Iran.
AUTHOR
[1] L.P. Belluce, The going up and going down theorems in MV-algebras and abelian groups, Journal of mathematical analysis and applications. Vol. 241 No. 4 (2000), pp. 92–106.
1
[2] L.C. Ciungu, A. Dvureˇcenskij and M. Hyˇcko, State BL-algebras, Soft Computing Vol. 15 No. 4 (2011), pp. 619–634.
2
[3] N. Constantinescu, On pseudo BL-algebras with internal state, Soft Computing. Vol. 16 No. 11 (2012), pp. 1915–1922.
3
[4] Z. Dehghani and F. Forouzesh, State filters in state residuated lattices, Categories and General Algebraic Structures with Applications. Vol. 10 No. 1 (2019), pp. 17–37.
4
[5] A. Di Nola and A. Dvurecenskij, On some classes of state-morphism MValgebras, Mathematica Slovaca. Vol. 59 No. 5 (2009a), pp. 517–534.
5
[6] A. Di Nola and A. Dvurecenskij, State-morphism MV-algebras, Annals of Pure and Applied Logic. Vol. 161 No. 2 (2009b), pp. 161–173.
6
[7] A. Dvurecenskij, J. Rachonek and D. Salounove, State operators on generalizations of fuzzy structures, Fuzzy sets and systems. Vol. 187 No. 1 (2012), pp. 58–76.
7
[8] T. Flaminio and F. Montagna, An algebraic approach to states on MV-algebras, in ‘EUSFLAT Conf.(2) (2007), pp. 201–206.
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[9] T. Flaminio and F. Montagna, MV-algebras with internal states and probabilistic fuzzy logics, International Journal of Approximate Reasoning. Vol. 50 No. 1 (2009), pp. 138–152.
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[10] N. Galatos, P. Jipsen and T. Kowalski, Residuated lattices: an algebraic glimpse at substructural logics, Vol. 151, Elsevier, (2007). [11] G. Georgescu and C. Muresan, Going up and lying over in congruence-modular algebras, Mathematica Slovaca. Vol. 69 No. 2 (2019), pp. 275–296.
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[12] P. He, X. Xin and Y. Yang, On state residuated lattices, Soft Computing. Vol. 19 No. 8 (2015), pp. 2083– 2094.
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[13] P. Jipsen and C. Tsinakis, A survey of residuated lattices in Ordered algebraic structures, Springer, (2002), pp. 19–56.
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[14] M. Kondo, Generalized state operators on residuated lattices, Soft Computing. Vol. 21 No. 20 (2017), pp. 6063–6071.
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[15] M. Kondo and M.F. Kawaguchi Some properties of generalized state operators on residuated lattices, IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL)’, IEEE, (2016) pp. 162–166.
14
[16] S. Rasouli, The going-up and going-down theorems in residuated lattices, Soft Computing. Vol. 23 No. 17 (2017), pp. 7621–7635.
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[17] S. Rasouli and B. Avvaz An investigation on boolean prime lters in BL-algebras, Soft Computing. Vol. 19 No. 10 (2015), pp. 2743–2750.
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[18] S. Rasouli and S. Zarin On residuated lattices with left and right internal state, Fuzzy Sets and Systems. Vol. 373 (2019), pp. 37–61.
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[19] M. Taheri, F. Khaksar Haghani and S. Rasouli, Simple, local and subdirectly irreducible state residuated lattices, (accepted by Revista de la union matematica argrntina on December 17, 2019)
18
ORIGINAL_ARTICLE
On the small intersection graph of submodules of a module
Let $M$ be a unitary left $R$-module, where $R$ is a (not necessarily commutative) ring with identity. The small intersection graph of nontrivial submodules of $M$, denoted by $\Gamma(M)$, is an undirected simple graph whose vertices are in one-to-one correspondence with all nontrivial submodules of $M$ and two distinct vertices are adjacent if and only if the intersection of corresponding submodules is a small submodule of $M$. In this paper, we investigate the fundamental properties of these graphs to relate the combinatorial properties of $\Gamma(M)$ to the algebraic properties of the module $M$. We determine the diameter and the girth of $\Gamma(M)$. We obtain some results for connectivity and planarity of these graphs. Moreover, we study orthogonal vertex, domination number and the conditions under which the graph $\Gamma(M)$ is complemented.
https://as.yazd.ac.ir/article_1936_6252b38beeed06188374ce696edc66c1.pdf
2021-02-01
117
130
10.22034/as.2020.1936
complemented graph
domination number
orthogonal vertex
Planar
small intersection graph
Lotf Ali
Mahdavi
l.a.mahdavi154@gmail.com
1
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
LEAD_AUTHOR
Yahya
Talebi
talebi@umz.ac.ir
2
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
AUTHOR
[1] S. Akbari, R. Nikandish, M. J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Alg. Appl., Vol. 12 No. 4 (2013), 1250200, 13 pp.
1
[2] S. Akbari, R. Nikandish, Some results on the intersection graph of ideals of matrix algebras, Linear Mult. Alg., Vol. 62 No. 2 (2014), pp. 195-206.
2
[3] S. Akbari, A. Tavallaee and S. Khalashi Ghezelahmad, Intersection graph of submodule of a module, J. Algebra Appl., Vol. 11 No. 1 (2012), 1250019, 8 pp.
3
[4] A. Amini, B. Amini, E. Momtahan and M.H. Shirdareh Haghighi, On a graph of ideals, Acta Math. Hungar., Vol. 134 No. 3 (2012), pp. 369-384.
4
[5] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, (1992).
5
[6] J.A. Beachy, Introductory Lectures on Rings and Modules, Cambridge University Press, London, (1999).
6
[7] J.A. Bondy and U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics 244, Springer, New York, (2008).
7
[8] J. Bosak, The graphs of semigroups, in Theory of Graphs and Application, (Academic Press, New York, 1964), pp. 119-125.
8
[9] I. Chakrabarty, S. Gosh, T.K. Mukherjee and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math., Vol. 309 (2009), pp. 5381-5392.
9
[10] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkauser Verlag, (2006).
10
[11] B. Csakany and G. Pollak. The graph of subgroups of a nite group, Czech Math. J., Vol. 19 (1969), pp. 241-247.
11
[12] S. Jafari and N. Jafari Rad, Planarity of intersection graphs of ideals of rings, Int. Electron. J. Algebra, Vol. 8 (2010), pp. 161-166.
12
[13] S.H. Jafari and N. Jafari Rad, Domination in the intersection graphs of ring and modules, Ital. J. Pure Appl. Math., Vol. 28 (2011), pp. 17-20.
13
[14] L.A. Mahdavi and Y. Talebi, Co-intersection graph of submodules of a module, J. Algebra Discrete Math., Vol. 21 No. 1 (2016), pp. 128-143.
14
[15] L.A. Mahdavi and Y. Talebi, Properties of co-intersection graph of submodules of a module, J. Prime Res. Math., Vol. 13 (2017), pp. 16-29.
15
[16] L.A. Mahdavi and Y. Talebi, Some results on the co-intersection graph of submodules of a module, Comment. Math. Univ. Carolin., Vol. 59 No.1 (2018), pp. 15-24.
16
[17] P. Malakooti Rad and L.A. Mahdavi, A note on the intersection graph of submodules of a module, J. Interdisciplinary Math., Vol. 22 No. 4 (2019), pp. 493-502.
17
[18] T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory, Siam, (1999).
18
[19] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, (1991).
19
[20] E. Yaraneri, Intersection graph of a module, J. Algebra Appl., Vol. 12 No. 5 (2013), 1250218, 30 pp.
20
[21] B. Zelinka, Intersection graphs of nite abelian groups, Czech Math. J., Vol. 25 No. 2 (1975), pp. 171-174.
21
ORIGINAL_ARTICLE
A new approach to smallness in hypermodules
In this paper, we extend the concept of small subhypermodules to all types of hypermodules and give nontrivial examples for this concept. As an application, we define and study lifting hypermodules via small subhypermodules.
https://as.yazd.ac.ir/article_1962_910efe4c70172e400e50e03dfbcc9aec.pdf
2021-02-01
131
145
10.22034/as.2020.1962
direct summand
lifting hypermodule
small subhypermodule
Ali Reza
Moniri Hamzekolaee
a.monirih@umz.ac.ir
1
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
LEAD_AUTHOR
Morteza
Norouzi
m.norouzi@ub.ac.ir
2
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran
AUTHOR
Violeta
Leoreanu-Fotea
violeta.fotea@uaic.ro
3
Faculty of Mathematics, University of Al.I. Cuza of Iasi, Iasi, Romania
AUTHOR
[1] S. M. Anvariyeh and B. Davvaz, Strongly transitive geometric spaces associated to hypermodules, J. Algebra, Vol. 322 (2009), pp. 1340–1359.
1
[2] S. M. Anvariyeh and B. Davvaz, On the heart of hypermodules, Math. Scand. Vol. 106 (2010), pp. 39–49.
2
[3] S. M. Anvariyeh and S. Mirvakili, Canonical (m, n)-ary hypermodules over Krasner (m, n)-ary hyperrings, Iran. J. Math. Sci. Inform. Vol. 7 No. 2 (2012), pp. 17–34.
3
[4] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules: Supplements and Projectivity in Module Theory Frontiers in Mathematics, Boston, Birkhuser, (2006).
4
[5] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, (1993).
5
[6] P. Corsini, V. Leoreanu-Fotea, Applications of Hyperstructure Theory, Kluwer Academic Publishers, (2003).
6
[7] B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. (2013).
7
[8] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, (2007).
8
[9] D. Keskin, On lifting modules, Comm. Algebra Vol. 28 No. 7 (2000), pp. 3427–3440.
9
[10] W. W. Leonard, Small modules, Proc. Amer. Math. Soc. Vol. 17 (1966), pp. 527–531.
10
[11] F. Marty, Sur une generalization de la notion de groupe, 8iem congres des Mathematiciens Scandinaves, Stockholm (1934), pp. 45–49.
11
[12] S. H. Mohamed and B. J. M¨uller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, (1990).
12
[13] A. R. Moniri Hamzekolaee and M. Norouzi, A hyperstructural approach to essentiality, Comm. Algebra Vol. 46 No. 11 (2018), pp. 4954–4964.
13
[14] B. Talaee, Small subhypermodules and their applications, Roman. J. Math. Comput. Soc. Vol. 3 No. 1 (2013), pp. 5–14.
14
[15] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, (1991).
15
[16] T. Vougiouklis, Hyperstructures and Their Representations Hadronic, press, Inc. (1994).
16
ORIGINAL_ARTICLE
Generalization of reduction and closure of ideals
Throughout this paper, all rings are commutative with identity and all modules are unital. Let $R$ be a ring and $M$ be an $R$-module. Then $M$ is called a multiplication module provided for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N=IM$. Also $M$ is said to be a comultiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $ N=(0:_MI)$. In this paper, we introduce the notions of reduction and coreduction of submodules, integral dependence, integral codependence, integral closure and $\Delta$-closure over multiplication and comultiplication modules.
https://as.yazd.ac.ir/article_1957_d406096da556d310456d35e0824642bd.pdf
2021-02-01
147
161
10.22034/as.2020.1957
Integral closure
integrally dependent
Multiplication modules
Reduction
Jafar
Azami
jafar.azami@gmail.com
1
Department of Mathematics, Faculty of Science, University of mohaghegh Ardabili, Ardabil
LEAD_AUTHOR
Maryam
khajepour
maryamkhajepour@uma.ac.ir
2
Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran
AUTHOR
[1] Z. Abd El-Bast and P. F. Smith, Multiplication modules , Comm. Algebra. 16(1988), no. 4, 755-779.
1
[2] M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, for associated primes of local cohomology modules, Comm. Algebra. 36, no. 12, (2008), 4620-4642.
2
[3] M. M. Ali, Residual submodules of multiplication modules, Beitr. Algebra Geom. 46(2) (2005),405-422.
3
[4] H. Ansari-Toroghy and F. Farshadifar, Product and dual product of submodules, Far East J. Math. Sci, 25, no. 3, (2007), 447-455.
4
[5] H. Ansari-Toroghy and F. Farshadifar, The dual notion of multiplication modules, Taiwanese J. Math, 11, no. 4, (2007), 1189-1201.
5
[6] R. Ameri,on the prime submodules of multiplication modules,The scientific world journal. no.27, (2003), 1715-1724.
6
[7] A. Barnard, Multiplication modules, Manuscripta Math. 71(1981), no. 1, 174-178.
7
[8] M.P. Brodmann and R.Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge University Press, Cambridge,1998.
8
[9] H. Matsumura, Commutative ring theory, Cambridge University press, Cambridge, UK, 1986.
9
[10] R. L. McCasland and M. E. Moore, On radicals of submodules of finitely generated modules, Canad. Math. Bull, 29 (1986) 37-39.
10
[11] A. G. Naoum and A.S. Mijbass, Weak cancellation modules, Kyungpook Math. J.,37(1997), 73-82.
11
[12] D. G. Northcott and D. Rees, Reductions of ideals in local rings , Proc. Cambridge Philos. Soc., 50(1954), 145-158.
12
[13] L. J. Ratliff, Jr., ∆ - closure of ideals and rings, Trans. Amer. Math. Soc., Volume 313, (1989), 221-247.
13
[14] Y. Tiras, Integral closure of an ideal relative to a module and ∆ - closure , Trans. Amer. Math. Soc.,21(1997), 381-386.
14
ORIGINAL_ARTICLE
Left $\phi$-biprojectivity of some classes of abstract Segal algebras
In this paper, we investigate left $\phi$-biprojectivity of Segal algebras and abstract Segal algebras. We show that for some abstract Segal algebras with some mild conditions left $\phi$-biprojectivity is equivalent with left $\phi$-contractibility. Also, we characterize left $\phi$-biprojectivity of a Segal algebra $S(G)$ in the terms of compactness of $G,$ where $G$ is a locally compact group. We introduce a class of abstract Segal algebras among Triangular Banach algebras. We show that some abstract Segal algebras related to triangular Banach algebras are not biprojective.
https://as.yazd.ac.ir/article_1960_8d6eda6c04bd1e10fb72afbd6a4e6fcf.pdf
2021-02-01
163
171
10.22034/as.2020.1960
Left $phi$-biprojective
Left $phi$-contractible
Locally compact group
Segal algebra
Triangular Banach algebra
Amir
Sahami
a.sahami@ilam.ac.ir
1
Department of Mathematics, Faculty of Basic Sciences, Ilam University P.O. Box 69315- 516 Ilam, Iran.
LEAD_AUTHOR
[1] M. Alaghmandan, R. Nasr Isfahani and M. Nemati, Character amenability and contractibility of abstract Segal algebras, Bull. Austral. Math. Soc, 82 (2010), 274-281.
1
[2] B. E. Forrest and L. E. Marcoux; Derivations of triangular Banach algebras, Indiana Univ. Math. J. 45 (1996), 441-462.
2
[3] B. E. Forrest and L. E. Marcoux; Weak amenability of triangular Banach algebras, Trans. Amer. Math. Math. J. 354 (2001), 1435-1452.
3
[4] F. Ghahramani and A. T. Lau, Weak amenability of certain classes of Banach algebra without bounded approximate identity, Math. Proc. Cambridge Philos. Soc 133 (2002), 357-371.
4
[5] Z. Hu, M. S. Monfared and T. Traynor, On character amenable Banach algebras, Studia Math. 193 (2009), 53-78.
5
[6] A. Jabbari, T. Mehdi Abad and M. Zaman Abadi, On φ-inner amenable Banach algebras, Colloq. Math. vol 122 (2011) 1-10.
6
[7] R. Nasr-Isfahani and S. Soltani Renani; Character contractibility of Banach algebras and homological properties of Banach modules, Stud. Math. 202(3), (2011), 205-225.
7
[8] H. Reiter; L 1 -algebras and Segal Algebras, Lecture Notes in Mathematics 231 (Springer, 1971).
8
[9] V. Runde, Lectures on amenability, Springer, New York, (2002). [10] A. Sahami On left φ-biprojectivity and left φ-biflatness of certain Banach algebras, U.P.B. Sci. Bull., Series A.Vol. 81, Iss. 4, 2019.
9
[11] S. S. Salimi, A. Mahmoodi, A. Sahami and M. Rostami, Left φ-biflatness and φ-biprojectivity of certain Banach algebras, Submitted.
10
[12] H. Samea, Essential amenability of abstract Segal algebras, Bull. Austral. Math. Soc, 79 (2009), 319-325.
11